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The b Matrix in Diffusion Tensor Echo-Planar ImagingJames Mattiello, Peter J Basser, Denis Le BihanIn diffusion tensor imaging (DTI) an effective diffusion tensorin each voxel is measured by using a set of diffusion-weightedimages (DWls) in which diffusion gradients are applied in amultiplicity of oblique directions. However, to estimate thediffusion tensor accurately, one must account for the effectsof all imaging and diffusion gradient pulses on each signalecho, which are embodied in the b matrix. For DTI to bepractical clinically, one must also acquire DWls rapidly andfree of motion artifacts, which is now possible with diffusion-weighted echo-planar imaging (DW-EPI). An analytical ex-pression or the b matrix of a general DW-EPI pulse sequenceis presented and then validated experimentally by measuringthe diffusion tensor in an isotropic phantom whose diffusivityis already known. The b matrix is written in a convenienttabular form as a sum of individual pair-wise contributionsarising from gradient pulses applied along parallel and per-pendicular directions. While the contributions from readoutand phase-encode gradient pulse trains are predicted to havea negligible effect on the echo, the contributions from otherimaging and diffusion gradient pulses applied in both paralleland orthogonal directions are shown to be significant in oursequence. In general, one must understand and account forthe multiplicity of interactions between gradient pulses andthe echo signal to ensure that diffusion tensor imaging isquantitative.Key words: MRI; b matrix; EPI; diffusion tensor.

INTRODUCTIONMR diffusion imaging (DI) 1-3) s a noninvasive in vivomethod to measure molecular diffusion of water in tis-sues, which has generated great scientific and clinicalinterest (4). Diffusion imaging (DI) consists of obtainingdiffusion-weighted images (DWI) directly or using themto calculate an apparent scalar diffusion constant (ADC)whose value can be displayed i n each voxel (5-7). How-ever, in anisotropic tissues, such as white matter (8) andskeletal muscle 9), the scalar ADC depends on the di-rection of diffusion sens itizing gradient and the tissueslocal fiber-tract direction. In these heterogeneous, aniso-tropic tissues, it is appropriate to characterize diffusivetransport of water by an effective diffusion tensor, D,rather than an ADC (10-14).

The practical importance of the effective diffusion ten-sor is that i t contains new and useful structural andphysiological information about tissues that was previ-ously unobtainable. Examples include the local fiber-

MRM 32292-300 1997)From the BiomedicalEngineering8 Instrumentation Program, National Cen-ter for Research Resources (J.M., P.J.B.). and Diagnostic Radiology De-partment, Warren G. Magnuson Clinical Center (D.L.B.), National Institutesof Health, Bethesda. Maryland.Address correspondence to: Denis Le Bihan, MD, Ph.D., Department ofResearch Imaging. Pharmacology, and Physiology, CEA, 4, place du Gen-eral-Leclerc. 91401 Orsay Cedex. France.Received October 4, 1995; revised May 15, 1996; accepted August 12,Copyright 1997 by Williams 8 WilkinsAll rights of reproduction n any form resewed.1996.0740-3194/97 $3.00

tract direction field, the principal diffusivities, and dif-fusion ellipsoids, which depict the mean-squareddiffusion distances of protons ( 1 5 ) . Moreover, one canderive from D quantitative, scalar MR parameters thatbehave like histological or physiological stains (16).These include measures of mean diffusivity (orTrace(D)),of diffusion anisotropy of fiber structure, andof fiber organization (16), all of which are rotationallyand translationally invariant, and thus are independentof fiber orientation (i.e., free of orientational artifacts)(15, 16).

Recently, we presented methods to estimate D from aseries of diffusion-weighted spin-echo spectra 1 ) andsubsequently, from two-dimensional Fourier trans-formed (2DFT) spin-echo images (18).Just as in diffusionimaging, where one estimates an apparent diffusion con-stant (ADC) from DWIs by using a ( scalar) b factor de-rived from each gradient pulse sequence (19),n diffu-sion tensor imaging (DTI), we estimate an effectivediffusion tensor (with six independent elements) fromDWIs by using a b matrix (with six independent ele-ments) derived from each pulse gradient sequence (20).Until recently, DWIs obtained by using 2DFT spin-echosequences suffered from long acquisition times, poor spa-tial resolution, and high susceptibility to motion arti-facts, al l of which impaired the quality in vivo DTI andimpeded its clinical implementation. However, diffu-sion-weighted echo-planar imaging (DW-EPI)overcomessome of these limitations (21 ) . With it, one can acquire adiffusion-weighted image i n milliseconds, without bulkmotion artifacts (22-24).

Previously, we presented an analytical expression forthe b matrix of a 2DFT spin-echo DW pulse sequence(18).Here we present the b matr ix for a DW-EPI sequencein a more convenient tabular form. The DW-EPI sequencepresents a more formidable challenge due to its multi-plicity and variety of applied gradient pulses. In partic-ular, the multiple phase-encode and read-out gradientpulses (applied during the collection of the echoes) couldhave a significant effect on the measured signal attenua-tion. By using pulse parameters appropriate for our ex-periment. we also calculate numerical expressions forthe b matrix for the measured DWIs. We use them both toestimate (statis tically) an effective diffusion tensor ofwater in each voxel of an isotropic phantom, from whichwe construct images of diffusion tensor elements, as wellas diffusion ellipsoids 15) n each voxel to demonstratethe val idity of this DTI protocol.

THEORYThe b Matrix of a General DW-EPI SequenceIn DI, one estimates a scalar apparent diffusion coeffi-cient (ADC) from the measured spin-echo intensity,

292

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The b Matrix in Diffusion Tensor EPI 293A( , using sequence is similar to that of a 2DFT spin-echo pulsesequence, the image period of the former consists of a

train of read-out and phase-encode gradient pulses. Thesignal generated by an EPI sequence consists of a series ofechoes, each occurring at the center of its correspondingread-out gradient. In the read direction, spins are com-pletely refocused during the first read-out gradient pulse,as in the 2DFT spin-echo pulse sequence. In the read andslice directions, the b matrix contributions of the prepa-ration (b,,,,) and imaging (him) periods to the b matrixmay be added, ( b = b p r e p him), as illustrated in Fig. 1.This additivity does not apply to the phase-encode di-rection, however, since spins remain out of focus at theend of the preparation period.A commonly used approximation is to calculate the bmatrix at the center of the k-space. This is partially guar-anteed by the constraint that we impose when we designthe DW-EPI sequence, (i.e., k(27) = 0) . Strictly, since theb matrix depends on all gradient pulses seen by thespins at a given time t each point of k-space should beassociated with a specific b matrix:

Above, 6 is a scalar that depends on the pulse sequence,and A ( 0 ) s the signal intensity with 6 = 0.

Analogously, in DTI we estimate the effective diffusiontensor, D, from the measured spin-echo, using (20):

Above, b , is a component of the symmetric b matrix, b;Dij is a component of the symmetric effective diffusiontensor, D; A(b ) is the echo intensity for a gradient se-quence whose b matrix is b , and A ( 0 ) s the echo intensityfor b = 0.The b matrix in Eq. [2] is calculated from thepulsed gradient sequence using 2 0 ) :

b =l k(t) 2H(t T)k(T))(k(f) 2 H ( t T)k(T))df[3alwhere

k(t) = y G t)dt ;[3bl

G t )= CAt),G,M, G, t ))T; k(27)= 0Above, y is the gyromagnetic ratio, 2 7 is the echo time,H t ) is the (Heaviside) unit-step function, and G t ) s thecolumn vector representing the DWI gradient pulse se-quence. Sequence parameters are always chosen so thatthe net phase accumulation vanishes in each direction(as in Eq. [3b]).

From Eq. 121, we see that the logarithm of the echoattenuation equals the sum of elements of the diffusiontensor, each of which is premultiplied by a correspond-ing element of the b matrix. Therefore, for each DWI, theb matrix provides the weights for each element of diffu-sion tensor that determine their contribution to the echoattenuation.

While for a diffusion tensor spectroscopic sequenceone can easily derive analytical expressions for the bmatrix by integrating Eq. [3] (20). for diffusion tensorimaging sequences, particularly for DW-EPI sequences,this approach is infeasible. Moreover, in DTI, one usuallyobtains many DWIs with different diffusion gradientstrengths and directions. A new b matrix must be calcu-lated for each DWI. Therefore, it is prudent to evaluateEq. [3] either symbolically (as Price and Kuchel did forspectroscopic sequences (25), or as we did for 2DFT spinecho DW imaging sequences (18)), r numerically.In calculating an analytical expression for the b matrixfor the spin-echo DW-EPI sequence shown in Fig. 1,wehave accounted for all gradient pulses that typicallyarise, including localization, crusher, and diffusion gra-dients; all of which are known to affect echo intensity(19,26,27).While the preparation period of an EPI pulse

where r is the position vector, and A(k) is the signal withTI, T2,nd proton density contribution. Considering thatdiffusion contrast is given by low spatial frequencies, theb matrix is usually calculated at the center of the k-space,so Eq. [4] is approximated by:

In the case of a conventional 2D-FT spin-echo pulsesequence, this means that we can calculate the b matrixwithout considering the phase-encoding gradient at thetop of the echo. However, in EPI this is no longer possi-ble, since multiple phase-encode gradient pulses are dis-persed throughout the readout period. Furthermore, thelarge number of large amplitude, short duration read-outgradient pulses that occur during the collection of theechoes could introduce cumulative errors into the calcu-lation of the b matrix if only the center of k-space wereused.

CALCULATING THE ELEMENTSOF THE b MATRIXIn obtaining an analytical expression for the b matrixfrom Eq. [3] , we synthesized a generalized DW-EPI gra-dient pulse sequence, G t), hown in Fig. 1, rom a libraryof sinusoidal and trapezoidal pulses, and integrated themby using Mathematica. The parameter values for eachsequence are given in the caption of Table 1. By specify-ing the plane of the image to be acquired, we also specifythe read, phase, and slice directions with respect to x-, -,and z laboratory frame. It is generally more convenient toexpress the b matrix elements in the coordinate frame of

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294 Mattiello et al .the image rather than in the laboratory frame. Therefore,we use the subscripts r , s, and p , corresponding to "read-out,'' "slice-select," and "phase-encode,'' rather than thesubscripts x,y, and z. However, i t is always straightfor-ward to transform between the image coordinate systemand the laboratory coordinate system.

EXPERIMENTAL METHODSFrom the analytical expressions in Eqs. [6 ]-[ ll] , we cal-culated the diagonal and off-diagonal elements of the bmatrix each measured by using parameters derived froman EPI pulse sequence that was originally proposed byTurner et al. for diffusion imaging (6, 7),but which weadapted to diffusion tensor imaging. We acquired coronalDWIs (64 X 64 pixels) of a water phantom in a glasssphere whose temperature was stable at 15C duringimage acquisit ion. Trapezoidal diffusion gradients wereapplied in seven non-collinear (oblique) directions (read,phase, slice, read and phase, phase and slice, read andslice, and read a nd phase and slice).' We acquired a totalof 1 1 2 images on a 4.7-T spectroscopy/imaging system(GE Omega, Fremont, CA). The diffusion gradients wereincreniented in 16 equal steps from 0 to 1.5 G/mm (150mT/m). The b matrix for each image was calculated off-line by using the imaging and diffusion gradient param-eters calculated from an EPI pulse sequence listed inTable 2.

Our imaging parameters were as follows: phase-encoderesolution (res)= 64, slice thickness = 2 mm, ramp time(rt)= 0.500 ms, read-out ramp time (rt r)= 0.200 ms, echotime ( 2 7 = TE) = 85.710 ms, and repetition time 73)15 s. The ramp time for the read-out and phase-encodegradients can be different from the ramp time for theother imaging gradients.Once numerical values of the b matrix were calculatedfrom the analytical expressions for each DWI, we usedmultivariate linear regression of Eq. [2 ] , as described inref. 20, o estimate in each voxel from the series ofDWIs.

RESULTS AND DISCUSSIONAnalytical Expressionsfor the b MatrixEquation [5] allows us to determine analytical expres-sions for the diagonal and off-diagonal elements of the bmatrix for the DW-EPI pulse sequence shown in Fig. 1.The evaluation of the b matrix is greatly simplified byobserving that the complicated double integral in Eq. [3]can be expressed as a sum of pair-wise interactions be-tween individual gradient pulses (18) whose terms rep-resent the product of two gradient amplitudes (G2/mm2)and a timing parameter (s ). For a DW-EPI sequence,some of these interactions are illustrated in Fig. 1 andexplained in detail in its caption. The analytical expres-sions for the b matrix represent the sum of contributionsarising from individual pair-wise interactions between

Technically, it is only necessary to apply diffusion gradients in six non-collinear directions.

FIG. 1. A general DW-EPI) sequence. The imaging and diffusiongradient intensities are a s follows: G, s a 90" slice-selectiongradient; G2 is a read-dephasing, phase-dephasing,or slice-refo-cusing gradient; G, s a diffusion gradient in the read, phase, orslice directions, respectively; G4are the crusher gradients in theread, phase, or slice directions, respectively;G, is the 180 lice-selection gradient;G, is the phase-encode gradient train; and G,is t h e read-out gradient train. The numbered boxes indicate pair-wise interactions that may exist between gradients in a typicalDW-EPI pulse sequence. They are between: 1 ) diffusion gradientsin the same direction (i.e., read and read, phase and phase, orslice and slice);2) diffusion gradients in different directions (i.e.,read and phase, read and slice, or phase and slice);3)diffusiongradients and imaging in t h e same direction (e.g., read-dephaseand read diffusion gradients, or 90 lice selection gradients andslice diffusion gradients); 4 diffusion gradients and imaging indifferent (orthogonal) directions (e.g., read-dephase and slicedif-fusion gradients, or phase crusher gradients and slice diffusiongradients);5) imaging gradients in the same direction (e.g., be-tween a pair of read crusher gradients, a 90" slice refocusinggradient and 90" slice selection gradient, or a read-dephase gra-dient and read-out gradients);6) imaging gradients in different(orthogonal) directions (e.g., read crusher gradients and slicecrusher gradients, read-dephase gradient and 180"slice selectiongradient);7) series of imaging gradients in the same direction(i.e.. the phase-encode gradient pulses); and 8) a seriesof imaginggradients in different (orthogonal) directions (i.e., read-out andphase-encode gradients). Only the interactions labeled1 3, and 5had been considered previously in diffusion imaging.individual gradient pulses in G ( t ) i.e., those in whichmost other gradients are set to zero). In this way, we areable to construct a table of all the possible interact ionsbetween one gradient pulse and all other relevant gradi-ent pulses occurring in the sequence, given in Table 1.Moreover, to evaluate any b matrix element, one simplysums up the appropriate elements of Table 1.The form ofTable 1 illustrates the fact that the b matrix can be ex-pressed as a sum of terms each of which represents anindividual pair-wise interaction between imaging or dif-fusion gradient pulses applied along the same or alongorthogonal directions (18).

Recognizing that the b matrix is a sum of pair-wiseinteractions between gradient pulses also greatly simpli-fies the process of evaluating the b matrix for a new ormodified imaging sequence. Once we have calculated thecontribution of one gradient pair (for example, betweentwo trapezoidal diffusion or crusher gradients), we donot have to recalculate them when we modify an existing

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The b Matrix in D i f f u s i o n Te n s o r EPI 295Table 1The Analytical Expression f o r b , and a Pictoral Representation of How Each of Its Terms Arise

This table shows al l possible pair-wise interactions between pulsed gradients n the general DW-EPI sequen ce shown in Fig. 1. All the gradient magnitudesand timing parameters are i l lustrated in cartoons of the gradient pulses. The top row indicates the gradient pulses that may be present in one gradientsequence, G, t ) ; he far-left co lumn indicates the gradient pulses that may b e present in another gradient sequenc e, G j t .To use this table to calcul ate thematrix for each seauenc e. dentifv al l those Du kes that do no t am ear in the sequence and set their gradient magnitudes to zero, then just add the remaining.terms in the table to obta in the b katr ix e lement b,, = b,,.sequence or analyze a new sequence. Therefore, many ofthe terms that we have already derived for the 2DFT spinecho DW imaging sequence also appear in the b-matrix ofthe DW-EPI sequence (18).

Sometimes, by inspection or design (e.g., by exploitingsymmetry principles), one can further simplify the formof the b matrix. First, interactions between a refocusedpulse and pulses applied previously or subsequently inorthogonal directions do not contribute to the off-diago-nal elements of b. For example, the 180 slice-selectgradient, as well as the diffusion and crusher gradientpairs applied in the slice-selection direction are effec-tively refocused in the spin echo sequence, and thereforeproduce no pair-wise interactions with the readout andphase-encode gradient pulse trains. Analytically, this isbecause the integrals of k, t)k,(t) as well as k, t)kAt) (inEq. 131) are unchanged by the application of these refo-cused pulses. In Table 1, these interactions contributezero to the b matrix. In addition, the potentially complexinteractions between read-out and phase-encode gradientpulse trains in the EPI sequence can be shown to have noeffect on b,,, during each of the periods b2 through b8 in

Fig. 2. During each of these periods, owing to the (oddand even) symmetry of these pulses there is no net con-tribution to the integral of k,[t) k,(t) (in Eq. 131). There-fore, only the first half of the first readout gradient pulseG,,)nd of the first phase-encode gradient [ G 6 , )contrib-ute to the b matrix element brp. In general, one canmitigate (and sometimes eliminate) the contribution ofany imaging gradient to the b matrix by refocusing it assoon as possible after it is applied. Clever pulse sequencedesign strategies that simplify the form of the b matrix arestrongly recommended and encouraged.

Using Table 1 to Evaluate the b MatrixFigure 1contains a DW-EPI sequence for which a generalb matrix is calculated using Eq. [3]. Since each element ofthis general b matrix is a sum of pair-wise interactionsbetween applied gradients, evaluating it just requiresidentifying which pair-wise interactions to retain andwhich to ignore. In Table 1, the Gi represent the pulsegradient magnitudes. The first index, k , indicates thetype of gradient pulse (e.g., slice-selection, phase-en-

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296 Matt ie l lo e t a l .

I

slice

FIG. 2. The phase-encode and read-out gradient pulse train for aseries of eight echoes.To calculate the b matrix elements b, andbPp we integrated up to the center of k-space (top of the echo) foreach read-out and phase-encode gradient, respectively. We seethat,by symmetry, contributions to b, from periodsb2 through b8vanish.code, etc.). The imaging and diffusion gradient intensi-ties are defined as follows: k = corresponds to a 90"slice-selection gradient; k = 2 corresponds to the read-dephasing, phase-dephasing, or slice-refocusing gradi-ents, respectively; k = 3 corresponds to the diffusiongradients in the read, phase, and slice directions, respec-tively; k = 4 corresponds to the crusher gradients in t heread, phase, and slice directions, respectively; k = 5corresponds to the 180 slice-selection gradient: k = 6corresponds to the phase-encode gradient train; and k =7 corresponds to the read-out gradient train. The secondindex, i, indicates the coordinate direction in which thatpulse is applied (i.e., the read, phase, or slice direction);the index m ndicates the gradient number (i.e., Is t, 2nd ,3rd. . . .). For gradient pulses lying along the same coor-dinate direction, i = j ; for gradient pulses lying alongdifferent coordinate directions, i # j .Table 2Parameters for a 4.7 T MR System

To determine the component b,, we first inspect thegradient sequences G,(t) and Gj(t) (e.g., in Fig. 1).Then weidentify all pulses present in Gj(t) depicted along the toprow of Table 1, and all pulses present in GJt) depictedalong the left column. All pulses shown in Table 1 thatdo not appear in G,(t) can be eliminated either by draw-ing a vertical line through them that extends down thecolumn or by setting their gradient magnitudes to zero.All pulses depicted in Table 1 not appearing in Gj(t) cansimilarly be eliminated either by drawing a horizontalline through them that extends across a row or by settingtheir gradient magnitudes to zero. Now that all gradientpulses have been accounted for, the b matrix element, b,is then obtained by taking the sum of all the remainingelements given in Table 1. By performing these steps foreach pair of gradient sequences, we can determine eachelement of the b matrix.

For instance, when we insert the pulse parametersgiven in Table 2 and in the Methods section in the ex-pressions for the components of the b matrix elementsgiven in Table 1, we obtain the following numericalexpressions:brr= 3.51 + 136.89G3,+ 2228.52G3; [6albpp= 3.19 + 130.63G3p+ 2228.52G3p2b,, = 1.76 58.5063, 2228.52G3:

[6bl[6cl

b, = b,, = 3.17 + 68.45G3, + 65.31G3,I6dl+ 2228.52G3,G3p

b, = b,, = -1.93 29.2563, + 68.4563,[6f2228.52G;3,63,

G,6, ms ti m s Glmm123

(31)4

(41)5

(32)6171

(42)

2.25002.0000

10.5000

2.50002.2500

0.20000.4400

1.37501.3750

21.480

37.48041.48045.23053.23065.23065.430

G s lGrdpGsrfGdr

Gcr

G P ~Gro

-0.2810.1290.1230.2480 to 0.9940 to 0.9510 to -0.9480.2010.192

-0.191

0.02440.489

The parameters used to calculateb matrix values for a DW-PI pulse sequence including the gradient strength and timing parameters used to calculate theb matrix values for the DW-EPI pulse sequence shown in Fig. 1. The k' gradient pulse strengths Gk),he gradient pulse timing parameter ak),nd the timeduring which the gradient pulses are turned on during the pulse sequences tc) are also defined.

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The b Matrix in Diffusion Tensor EPI 297These expressions show how the b matrix elements

(given in (s/mm)) depend on the diffusion gradientstrengths, G,,, G,,,, and G,, (G/mm) n our experiment. InDTI we acquire a series of DWIs by varying the strengthand sign of the three diffusion gradients from which weobtain estimates of D (20). (Above, G,, is assumed to benegative: see Table 2). In Eqs. [6a]-[6fl, all constant termsarise solely from interactions between imaging gradients.All linear terms in G,,, GJp, or G,, arise solely frominteractions between imaging and diffusion gradients.Finally, all quadratic terms in G,,, GBp,or G,, arise solelyfrom interactions between diffusion gradients. The diag-onal elements of the b matrix (i.e., b b,,, b,,) containcontributions from interactions between gradient pulsesapplied along the same direction, whereas the off-diago-nal elements of the b matrix (i.e., b b,,,, b,) containcontributions from interactions between gradient pulsesapplied along orthogonal directions. The terms that arequadratic in only one of the diffusion gradients (i.e.,G,:,G,,,, and G result from interactions between diffu-sion gradients applied along the same direction. Theseare the well-known Stejskal and Tanner terms used indiffusion spectroscopy and (by many) in diffusion imag-ing to estimate ADCs. The other quadratic terms (i,e,,G3rC3p. G3pG3s, nd G3,G3s) represent interactions be-tween diffusion gradients applied in orthogonal direc-tions (20).We consider all terms in Eqs. [6a]-[6fl that donot depend on products of diffusion gradients, (i.e., arenot of the form G,jG,j) cross terms because they de-pend on imaging gradients. While the effect that gradi-ents applied in the same directions have on the echomagnitude had previously been considered in diffusionimaging studies (19, 26, 27), the effect that gradientsapplied in orthogonal directions have on the echo mag-nitude had not until recently (18, 20). Some of the pair-wise interactions between gradients pulses that give riseto cross terms are illustrated in Fig. 1. In summary, dif-fusion tensor imaging provides a more general frame-work than diffusion imaging for treating the contributionthat each gradient pulse has on echo attenuation in dif-fusion weighted sequences.

To assess the importance of the applied diffusion gra-dients, b-matrices below were calculated using Eqs. (6al-[Sfl . When no diffusion gradients are applied (i.e., G,, =G S p = G,, = 0.0 G/mm), the b matrix b G,, G3p,G,,),written as a function of diffusion gradient amplitudes,becomes,

3.51 3.16 1.933.16 3.19 1.841.93 1.84 1.76

When G,, = 0.2 G/mm, the b-matrix becomes:120.03 16.23 7.78

16.23 3.19 1.847.78 1.84 1.76

When G,, = G,, = 0.2 G/mm, the b matrix becomes:120.03 16.23 95.05

16.23 3.19 7.6995.05 7.69 79.2

Similarly, when G,,. = G3p= G,, = 0.2 G/rnm, then the bmatrix becomes:

120.03 119.06 95.05119.06 118.45 94.5195.05 94.51 79.2

Eqs. [7]-[10] represent a typical range of b matrix valuesfor a clinical EPI system. If we recall that the b matrixelements are weighting factors calculated for each DWIthat are used to estimate the diffusion tensor elements,then we can appreciate the enormous range of relativeweights of different elements of the diffusion tensor canbe subjected to (both diagonal and off-diagonal ele-ments).By inspecting Eqs. [6]-[lo], we see that cross-terms aresignificant in this DWI sequence. For example, in Eq. [8]where diffusion gradients are applied only in the rdirection, one might expect that all b matrix elementsother than b, would vanish. Yet all other elements of theb matrix are non-zero, and two are as high as 10 of b,.We can assess the significance of cross-terms in thisDW-EPI sequence by evaluating the error made in esti-mating the effective diffusion tensor when we ignorethem. Figure 3 shows the fractional error in b, Abr,/brr,that was calculated from Eq. [6a] plotted against applied

7EPI data

0 0.5 1 1.5 2 2.5

FIG. 3.The fractional error in b AbJb (from Eq. [Sa]),plottedagainst the applied diffusion gradient strength,Gdr When Gdr =0. he percentage error in b is 100%.The error drops off to about20% for Gdr = 0.2 G/mm. which is typical for the peak gradientstrength in a clinical EPI system, and is still about 5 for Gdr = 2.0G/mm.

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298 Mattiello et al.diffusion gradient strength, GI,..When G,, = 0 , the per-centage error in b, is 100%. The error drops off to about20 for G,, = 0.2 G/mm (a typical peak gradient strengthfor a clinical EPI system) and to about 5 for G,, = 2.0G/mm Since we can easily show from Eq. [2] that thepercentage error in b, equals the percentage error i n theestimated D,, we can see immediately that omitting theeffect of imaging gradients on the echo attenuation pro-duces a significant error in the estimated diffusion ten-sor.

However, ignoring cross terms introduces additionaldeleterious errors. Even when a diffusion gradient isapplied only along one direction, all diagonal and off-diagonal elements of the b matrix are still significantlydifferent from zero. Above, we see this is true when thediffusion gradient is zero (as in Eq. 171) or is large (as inEq. [el).This means that each element of D potentiallycan contribute to the observed signal attenuation. How-ever, when one uses the scalar model of diffusive trans-port (Eq. [I]) ather than the tensor model (Eq. [ 2 ] ) , onetacitly assumes that the signal attenuation results en-tirely from the action of gradients applied in the direc-tion in which the diffusion gradient is applied.The importance of imaging Gradients in DW-EPIAvram and Cooks 5) and Turner and LeBihan (6) sug-gested that the read-out gradients in the imaging phase ofthe EPI pulse sequence have a negligible effect on thesignal attenuation due to diffusion. We can assess thisassertion quantitatively by using the imaging sequenceshown in Fig. 1 he imaging parameters given in Table 2,and the b matrix formulae given in Eqs. [6a]-[6fl. In theread direction, the contribution to the b matrix during theimage period is brfliml 0.082 s/mmZ,which is negligi-ble. But if we treat each read-out gradient separately, wesee that the read-out gradient train provides a small butsignificant contribution to brr.The effect of all readoutgradients is 2 . 3 of b, when th e diffusion gradient G is0 0 G/mm and 0.07% of b, when G3,.was 0.2 G/mm.Interestingly, because of the symmetry of the read-outand phase-encode pulses (discussed above), the interac-tion between these two gradient pulse trains is negligible.It is only 0.004 s/mm2, which is less than 0.1% of b,.

Although ind ividual phase-encode gradient pulses aresmall, their integrated effect may still be significant. InEq. [3] , the contribution to the b matrix is not simplyadditive because the cumulative phase shifts are thenintegrated a second time. Even though its contribution tob,, is 0.121 s/mm2 , which is negligible, the contributionof the phase-dephasing gradient is not. Its contributionwas 3 . 5 of bI,, when the diffusion gradient strength was0.0 G/mm, and only 0.1% of b,, when the diffusiongradient strength was 0 .2 G/mm.

Validating t h e b MatrixIn principle, in an isotropic medium such as water, alloff-diagonal elements of the diffusion tensor vanish andthe three diagonal elements are equal to the scalar diffu-

sivity, Do, i.e.,

D = D o I = D , , 0 1 0 [111

where I is the identity tensor, and Do is the scalar self-diffusion constant at the temperature of the experiment(e.g., see ref. 20).

To validate the calculated b matrices, we estimate D ofwater from a series of DWIs, an d then assess whether theestimated D is isotropic , (Lee, s of the form given in Eq.[I l l ) . If our b matrix elements were calculated incor-rectly, then we would expect the off-diagonal elements ofD to be significantly different from zero, and the diagonalelements of D to be significantly different from eachother, as well as different fromDo. This is because a givenpercentage error in the b matrix element produces thesame percentage error in the corresponding element ofthe (statistically) estimated diffusion tensor element butof an opposite sign (16). Therefore, an underestimated bmatrix element should produce an overestimate in thecorresponding diffusion tensor element, and visa versa.Therefore, errors in b should cause errors in D, whichshould make the estimated D for water deviate fromisotropy.

In the Methods section above, we explained how weestimate D in each voxel from a set of DW-EPIs by usingthe analytical expression for the b matrix that we evalu-ate numerically for each DWI. Figure 4 shows images ofthe six independent components of D for a water phan-tom obtained by using th is method. Images of each of thethree diagonal components D.xx, Dyyr and 0 J have ahigh but uniform intensity, while images of the threeoff-diagonal components Dx,,, Dxz, and D,,J are all at thelevel of background noise, indicating they are not statis-tically significant. Interestingly, they also show an edgeenhancement at the interface between the water sampleand the glass container. This artifact is caused by localgradients produced at the watertglass interface owing totheir differences in magnetic susceptibility. Susceptibil-

1 1 1

FIG. 4. Images of each of the six independent components of theestimated effective diffusion tensor,D, derived from DW-EPI for awater phantom. Top row (left to right): images of the diagonalcomponents of D Dxm0, and D bottom row (left to right):images of the off-diagonal componentsof D DXyOx, and DYJ.

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The b Matrix in Diffusion Tensor EPI 299ity gradients are, of course, not explicitly accounted forin the calculation of the b matrix.

From a 16 X 20 pixel region of interest (ROI) in thewater phantom, the mean effective diffusion tensor andits covariance were measured:

i.675 2 0.018 0.011 * 0.007 0.005 ? 0.0070.011 2 0.007 1.680* 0.014 0.016 2 0.0070.005 2 0.007 0.016 ? 0.007 1.665 t 0.0131 0 - ~mz/s [121The value of the diffusion coefficient of water at 15C haspreviously been reported as Do = 1.69 2 0.02 Xmm2/s (28, 291. The diagonal elements of D that wemeasured in Eq. [I21 compare extremely well with Do.Moreover, the off-diagonal elements of D are negligible.D,, is statistically indistinguishable from zero, while D X yand Dyz are significantly different from zero but stillnegligibly small. Overall, Eq. [ 1 2 ] has a form consistentwith that of an isotropic tensor.

To assess the degree of spatial variability or heteroge-neity, as well as the degree of anisotropy of the estimateddiffusion tensors in each voxel, we also constructed adiffusion ellipsoid image (see Fig. 5), as described previ-ously (15) for the 16 x 20 pixel ROI. As expected fromboth the isotropic form of the estimated D, and the smallvariances reported in Eq. [ 1 2 ] , the ellipsoids are allspherical with diameters that are uniform from voxel tovoxel. Their shape and size indicate that there is nopreferred direction for diffusion, and that there is virtu-ally no variability or heterogeneity in the measured dif-fusion coefficient of water in the phantom, which wewould have expected if we made a systematic error inaccounting for imaging gradients. If the diagonal ele-ments were not all similar, or if the off-diagonal compo-nents were significantly different from zero, thesespheres then would appear as prolate ellipsoids, withtheir polar axes aligned. These control studies demon-strate that we have made no systematic errors in calcu-lating the diagonal and off-diagonal elements of the b

FIG.5. Diffusion ellipsoid mage. Ellipsoids are constructed in eachvoxel of an ROI from D estimated in each voxel whose compo-nents are shown in Fig. 4. The spherical ellipsoids are consistentwith water being isotropic , indicating hat correctb matrix values

matrix from the gradient pulse sequence, and that weestimated the effective diffusion from the measured ech-oes properly, as well. Performing a calibration study likethis one is highly recommended before performing DTIon tissues or other media.

CONCLUSIONSIn arriving at an analytical expression for the diagonaland off-diagonal elements of the b matrix for a DW-EPIpulse sequence, we have taken into account the contri-butions arising from all applied imaging and diffusiongradients (i.e., applied along parallel and perpendiculardirections]. These analytical expressions permit one toevaluate the b matrix for a general DW-EPI sequenceefficiently, off-line. They also permit one to estimatestatistically the diagonal and off-diagonal elements of theeffective diffusion tensor, D, accurately and efficientlyfrom a series of DW-EPIs. This innovation has facilitatedthe successful implementation of high-resolution, high-quality, quantitative DTI (301, and may facilitate the de-velopment of quantitative DT-MR microscopy (e.g., ref.311,where large imaging gradients are required to attainhigh spatial resolution.

ACKNOWLEDGMENTSThis work was performed at the NIH In Vivo NMR ResearchCenter. The authors thank Alan Olson, Peter Jezzard, RobertTurner, Geoff Sobering, and Scott Chesnick for technical sup-port; and Barry Bowman, Brad Roth, Car10 Pierpaoli, a n d Z & RBigio for editing this manuscript.

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Announcements of MeetingsMagnetic Resonance Imaging of the Brain, Spine an d Musculoskeletal System, spon-sored by the University of California, San Diego, School of Medicine, will be held onMarch 2-7, 1997, at the Hotel Del Coronado, Coronado California. This course isdesigned for physicians and a llied heal th personnel and will cover the basic concepts ofMR imaging and techn iques, such as MR angiography, fat suppression MR imaging, andfast spin-echo sequences. Information from Ryals & Associates, Inc., P.O. Box 1925,Roswell, Georgia 30077-1925, tel (770) 641-9773, fax (770) 552-9859, email:Webmaster@Ryalsmeet.com, World Wide Web: http://www.ryalsmeet.com4th Annual Update in General Diagnostic Imaging: Breast, Abdominal and Neuroradi-ology Imaging, sponsored by the University of Chicago, Department of Radiology, willbe held on March 10-14, 1997, at the Breakers Resort Hotel, Palm Beach, Florida. Thisprogram will include a 4-hour mammography screening workshop, interventionalbreast procedures, screening for breast cancer, mammographic follow-up, imaging ofcirrhosis, GU radiology, biliary disease, helical CT, interventional, transrectal andprostatic US, hydrocephalus, MS and white matter lesions, posterior fossa abnormali-ties. Information from Ryals & Associates, Inc., P.O. Box 1925, Roswell, Georgia 30077-1925, tel (770) 641-9773, fax (770) 552-9859, email: Webmaster@Ryalsmeet.com,WorldWide Web: http://www.ryalsmeet.comMinimally Invasive Therapy of the Brain, sponsored by the American Association ofPhysician Specialists, and the International Institute for Continuing Medical Educa-tion, will be held on March 14-16, 1997, at th e Ritz-Carlton Hotel, Marina Del Rey,California. This 2 day program is designed to provide neuroradiologists, neurologists,and radiologists with a special interest in minimally invasive therapy, a review ofcurrent approaches to brain tumors, vascular disease, and movement disorders usingminimally invasive techniques. Information from Ryals & Associates, Inc., P.O. Box1925, Roswell, Georgia 30077-1925, tel (770) 641-9773, fax (770) 552-9859, email:Webmaster@Ryalsmeet .com,World Wide Web: http://www.ryalsmeet.com